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 Title: Inequalities for the differential subordinates of Martingales, harmonic functions and Ito processes Author(s): Choi, Changsun Doctoral Committee Chair(s): Ruan, Zhong-Jin Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: In Chapter 1 we sharpen Burkholder's inequality $\mu(\vert v\vert\geq1)\leq2\Vert u\Vert\sb1$ for two harmonic functions u and v by adjoining an extra assumption. That is, we prove the weak-type inequality $\mu(\vert v\vert\geq1)\leq K\Vert u\Vert\sb1$ under the assumptions that $\vert v(\xi)\vert\leq\vert u(\xi)\vert, \vert\nabla v\vert\leq\vert\nabla u\vert$ and the extra assumption that $\nabla u\cdot\nabla v$ = 0. Here $\mu$ is the harmonic measure with respect to $\xi$ and the constant 1 $In Chapter 2 we get norm inequalities. Let ($\Omega,{\cal F},P$) be a probability space with filtration (${\cal F}\sb{n}).$Let f be a nonnegative submartingale and g be an adapted sequence. Let d be the difference sequence of f and e of$g{:} f\sb{n}=\sum\limits\sbsp{k=0}{n}\ d\sb{k}$and$g\sb{n}=\sum\limits\sbsp{k=0}{n}\ e\sb{k}, n\ge 0.$We prove$\Vert g\Vert\sb{p}\le(r-1)\Vert f\Vert\sb{p}$under the assumption that$\vert e\sb{n}\vert\le\vert d\sb{n}\vert$for$n\ge 0$and$\vert{\rm I\!E}(e\sb{n}\ \mid\ {\cal F}\sb{n-1})\vert\le\alpha\vert{\rm I\!E}(d\sb{n}\ \mid\ {\cal F}\sb{n-1})\vert$for$n\ge 1.$Here 0$\le\alpha\le$1, 1$ Issue Date: 1995 Type: Text Language: English URI: http://hdl.handle.net/2142/19178 Rights Information: Copyright 1995 Choi, Changsun Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9624313 OCLC Identifier: (UMI)AAI9624313
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